In this chapter, we consider commutative groups and one of their most important applications in every day life. At the end of this chapter we present the Diffie-Hellman key exchange that is, for example, used when your web browser establishes a secure (https) connection with a web server.
We bring together together many of the topics from chapters 1, 2, and 3. The sets \(\Z_n\) and \(\Z^\otimes_n\) show up again, in particular the sets \(\Z^\otimes_p\) where \(p\) is a prime number will be of interest. We introduce new operations on these sets and revisit exponentiation. Finally, we apply these in real world encryption algorithms.
One of the most familiar examples of a commutative group is the set of integers with the addition operation. There is a wide variety of groups that find applications in a multitude of fields. In addition to their application in cryptography, groups are used to describe symmetries of objects in physics and chemistry.
In Chapter 13, we introduce binary operations and properties of binary operations. We give the definition of a commutative group and some examples of commutative groups in Chapter 14. As mentioned before, the \(\fmod\) operation will become important to us. We give some more applications of \(\fmod\) and then show how the sets \(\Z_n\) and \(\Z^\otimes_n\) together with operations based on \(\fmod\) and addition or multiplication, respectively, give us infinitely many groups. We present two families of groups whose operations are modular addition and modular multiplication, respectively. Within these groups, we examine groups that are generated by one element in Chapter 15 and show how they are used in public key crypto systems in Chapter 16.
